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Employing the identity (a+b)² = a² + 2ab + b², where a and b are any real numbers.

Substituting  a and b with a and x/y, respectively:

(a + x/y)² = a² + 2a(x/y) + (x/y)²

(a + x/y)² = (a + x/y)(a + x/y)

= a² + ax/y + ax/y + x²/y²

= a² + 2ax/y + x²/y²

add a² to both sides of the equation:

a² + (a + x/y)² = a² + 2ax/y + x²/y² + a²

a² + (a + x/y)² = (a² + x²/y²) + 2ax/y + a²

xy = a² substitute a² for xy in the last term:

a² + (a + x/y)² = (a² + x²/y²) + 2axy/xy + a²

a² + (a + x/y)² = (a² + x²/y²) + 2a + a²

We have that  x²/y² = (xy)/(y²) = a/y, and substituting this back into the equation:

a² + (a + x/y)² = (a² + a/y + a/y) + 2a + a²

= 2a² + (2a/x) + (2a/y)

A real number is described as a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

Therefore from the mathematical expression shown above, we have shown that:

a² + (a + x/y)² = 2a² + (2a/x) + (2a/y)

This is the mathematical expression we wanted to prove.

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ΔQRS ≅ ΔTUV and ΔTUV ≅ ΔXYZ then ΔXYZ  ≅ ΔQRS by Transitive Property.

ΔEFG ≅ ΔJKL then ΔJKL ≅ ΔEFG using Symmetric Property.

The transitive property of congruence states that “ if two shapes are congruent to the third shape, then all the shapes are congruent to each other.

This is the transitive property at work: if a=b and b=c, then a=c.

First, ΔQRS ≅ ΔTUV and ΔTUV ≅ ΔXYZ

The property of Transitivity shows for any angle <A, <B and <C.

So, <A = <B and <B = <C then <A = <C

Second, The property of Symmetry shows or any angle <A, <B

if  <A≅ <B then <B≅ <A

So, ΔEFG ≅ ΔJKL then ΔJKL ≅ ΔEFG using Symmetric Property.

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Answer:

See below.

Step-by-step explanation:

We are asked to simplify the expression provided.

An easy way to simplify this expression is by Combining Like Terms.

Combining Like Terms means adding terms that are similar, whether that's natural numbers, or variables.

For example, 8x + 9x are like terms since they both have the same variables. There's no exponents or any other variables that make these terms different from each other.

Using what we've learned, let's combine like terms for our problem now.

Combine Like Terms:

[tex]7h+6.3d-15+4d-3.4h[/tex]

[tex]7h - 3.4h = 3.6h[/tex]

[tex]6.3d + 4d = 10.3d[/tex]

-15 has no like terms.

We should have;

[tex]10.3d-15+3.6h[/tex]

This will be our final answer. There's no more like terms to combine and we can't simplify this expression further.

The number line graph A is correct, thus, option C is true.

In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.

Given, an inequality x ≥ -5.

The inequality x ≥ -5 means that x is greater than or equal to -5.

In other words, any value of x that is equal to or greater than -5 satisfies the inequality.

For example, x = -5, x = 0, x = 10, and x = 100 all satisfy the inequality x ≥ -5 because they are greater than or equal to -5.

On a number line, the solution set for this inequality would include all the values of x to the right of -5, including -5 itself,

because the dot at -5 would be filled in to indicate that it is included in the solution set.

The part of the number line to the left of -5 would not be included because those values of x are less than -5, which does not satisfy the inequality.

Therefore, the Option C is correct.

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Answer: N′(−7, −2), M′(−4, −1), O′(−7, −5)

Step-by-step explanation:

im right and youre wromng

Answer:

4 cats

Step-by-step explanation:

Let b represent birds and c represent cats

b = c + 3

c + b = 11

c + (c + 3) = 11

2c + 3 = 11

2c = 8

c = 4.

John has 4 cats

b = (4) + 3

b = 7

John has 7 birds

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Answer: The answer is the letter B

Step-by-step explanation:

Mean is the overall average of numbers or books and range is the length from greatest to least

Answer:

x = 5.5

Step-by-step explanation:

2x + 3 - 4 = 10

2x - 1 = 10

2x = 11

x = 5.5

Let's check

2(5.5) + 3 - 4 = 10

11 + 3 - 4  = 10

14 - 4 = 10

10 = 10

So, x = 5.5 is the correct answer.

Answer: x = 5.5

Step-by-step explanation:

To solve, we will isolate the variable x.

      Given:

             2x + 3 - 4 = 10

      Combine like terms on the left side:

             2x - 1 = 10

      Add 1 to both sides of the equation:

             2x = 11

      Divide both sides of the equation by 2:

             x = 5.5

The value of a discriminant determines the kind of roots in a quadratic equation, and the sign of a discriminant indicates whether the roots were real or complex, unique or recurring.

Definitions: x ax2 + bx + c = 0 is a quadratic equation, which is a 2nd quadratic formula in a single variable. a 0. It has at most one solution since it is a 2nd quadratic problem, which is guaranteed by the algebraic basic theorem. The solution could be straightforward or difficult.

To put it another way, if a twice the square of a expression that comes after b plus b times the same expression never squared plus c equals 0.

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(5[tex]\sqrt{3}[/tex])^2

= (25 x 3)

= 75

(2[tex]\sqrt{5}[/tex])^2

= (4 x 5)

= 20

The domain of a function is all the possible input values for which the function is defined.

Function is a block of code that performs a specific task. It is a reusable piece of code that can be called from other parts of the code, reducing the amount of code that needs to be written and making the code easier to read and maintain. Functions can take in parameters and return values, allowing for increased flexibility and reusability. Functions can also be used to break up large programs into smaller, more manageable pieces.

In this case, the reasonable domain for the function f(x) = 4x³-50x²+150x would be all real numbers. This is because the function is a polynomial, which is continuous, meaning that it is defined for all real numbers.

The input of the function is a width, and the output is a volume. Therefore, the domain of the function should include all reasonable widths that could produce a volume. This means that the domain should include all positive real numbers, as any negative value would produce a negative volume, which is not reasonable. Thus, the reasonable domain for the function f(x) = 4x³-50x²+150x is all positive real numbers.

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Answer:

[tex]\cos \dfrac{\theta}{2}(1-\cos \theta)=\sin \dfrac{\theta}{2} \sin \theta[/tex]

Step-by-step explanation:

[tex]\textsf{To show that}\;\;\cos \dfrac{\theta}{2}(1-\cos \theta)=\sin \dfrac{\theta}{2} \sin \theta:[/tex]

[tex]\textsf{Let}\;\;u=\dfrac{\theta}{2} \implies 2u=\theta[/tex]

Therefore:

[tex]\implies \cos \dfrac{\theta}{2}(1-\cos \theta)=\cos u(1-\cos 2u)[/tex]

 cos(A±B) = cosA cosB ∓ sinA sinB

 cos(2θ) = cos²θ - sin²θ

 cos(2θ) = 2cos²θ - 1

 cos(2θ) = 1 - 2sin²θ

Use the cos double angle identity cos(2θ) = 1 - 2sin²θ to rewrite cos(2u) in terms of sin:

[tex]\begin{aligned}\implies \cos \dfrac{\theta}{2}(1-\cos \theta)&=\cos u(1-\cos 2u)\\&=\cos u(1-(1-2 \sin^2 u))\end{aligned}[/tex]

Simplify:

[tex]\begin{aligned}\implies \cos \dfrac{\theta}{2}(1- \cos \theta)&=\cos u(1- \cos 2u)\\&=\cos u(1-(1-2 \sin^2 u))\\&=\cos u(1-1+2 \sin^2 u)\\&= \cos u(2 \sin^2 u)\\&=2 \sin u \cos u \sin u\end{aligned}[/tex]

 sin(A±B) = sinAcosB ± cosAsinB

 sin(2θ) = 2sinθcosθ

Using the sine double angle identity to rewrite 2sin(u)cos(u) as sin(2u):

[tex]\begin{aligned}\implies \cos \dfrac{\theta}{2}(1-\cos \theta)&=\cos u(1- \cos 2u)\\&=\cos u(1-(1-2 \sin^2 u))\\&= \cos u(1-1+2 \sin^2 u)\\&= \cos u(2 \sin^2 u)\\&=2 \sin u \cos u \sin u\\&= \sin(2u) \sin u\end{aligned}[/tex]

Finally, substitute u = θ/2 back in:

[tex]\begin{aligned}\implies \cos \dfrac{\theta}{2}(1-\cos \theta)&=\cos u(1-\cos 2u)\\&=\cos u(1-(1-2 \sin^2 u))\\&=\cos u(1-1+2 \sin^2 u)\\&= \cos u(2\sin^2 u)\\&=2\sin u\cos u \sin u\\&=\sin(2u) \sin u\\&=\sin\left(2 \cdot \dfrac{\theta}{2}\right) \sin \dfrac{\theta}{2}\\&=\sin \theta \sin \dfrac{\theta}{2}\\&=\sin \dfrac{\theta}{2} \sin \theta \end{aligned}[/tex]

If you don't want to substitute u = θ/2, the full calculation using the same double angle identities is:

[tex]\begin{aligned}\implies\cos\dfrac{\theta}{2}(1-\cos\theta)&=\cos\dfrac{\theta}{2}\left(1-\left(1-2\sin^2\dfrac{\theta}{2}\right)\right)\\\\&=\cos\dfrac{\theta}{2}\left(1-1+2\sin^2\dfrac{\theta}{2}\right)\\\\&=\cos\dfrac{\theta}{2}\left(2\sin^2\dfrac{\theta}{2}\right)\\\\&=2\sin \dfrac{\theta}{2}\cos \dfrac{\theta}{2} \sin \dfrac{\theta}{2}\\\\&=\sin\left(2\cdot\dfrac{\theta}{2}\right)\sin\dfrac{\theta}{2}\\\\&=\sin\theta\sin\dfrac{\theta}{2}\\\\&=\sin\dfrac{\theta}{2}\sin\theta \end{aligned}[/tex]

Based on the four options, Brianna should choose to invest her summer earnings in the Account 4 (compounded continuously) because interest is at highest in the account after like 3 years.

Interest is higher in compounded continuously because it is calculated more frequently than in other compounding methods, such as annually or semi-annually. In continuous compounding, interest is calculated and added to the principal an infinite number of times over a given period, which results in a higher effective interest rate than other compounding methods.

This means that the amount of interest earned on an investment or loan increases more rapidly, which can result in greater returns for investors or higher costs for borrowers. Continuous compounding is often used in financial markets where small changes in interest rates can have significant impacts on returns, such as in the case of short-term bonds or options trading.

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The angle measures are given as follows:

Two angles are called supplementary when the sum of their measures is of 180º.

On a trapezoid, consecutive interior angles are supplementary, hence the pairs of supplementary angles are given as follows:

Hence the value of y is obtained as follows:

y + 145 = 180

y = 35º.

(as the two angles are supplementary we know that the sum of their measures is of 180º).

The value of x is obtained as follows:

x + 62 = 180

x = 118º.

(as the two angles are supplementary we know that the sum of their measures is of 180º).

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If the bacteria in a dish that contains 2 bacteria quadruple every 16 minutes can be represented by the expression 2×4t/16, an equivalent expression for the number of bacteria in the dish for any time, t, in minutes is B. 8t/16.

An equivalent expression depicts two algebraic expressions that have the same value when the values for the variables are plugged in.

An algebraic expression is the combination of variables, values, constants, and numbers using mathematical operands without the equal symbol (=).

2×4t/16 = 8t/16

2×4t/16 ≠ 4t8

2×4t/16 ≠ 2×2t8

2×4t/16 ≠ 4×2t16

Thus, 2×4t/16 = 8t/16 as equivalent expressions.

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They are intersecting

a) Not parallel because they share different slopes

b) Not perpendicular, they dont intersect creating a right angle. This is because neither of the slopes are a reciprocal of each other.

Answer:

Let's denote the speed of the second car as x (in miles per hour). Then the speed of the first car is 12 miles per hour faster, which is x + 12.

We know that both cars traveled the same amount of time, so we can use the formula:

distance = speed × time

For the first car, we have:

234 = (x + 12) × time

For the second car, we have:

180 = x × time

We want to find the speeds of both cars, so we need to solve for x and x + 12. We can start by solving for time in both equations:

time = 234 / (x + 12)

time = 180 / x

Since both expressions are equal to time, we can set them equal to each other:

234 / (x + 12) = 180 / x

To solve for x, we can cross-multiply and simplify:

234x = 180(x + 12)

234x = 180x + 2160

54x = 2160

x = 40

Therefore, the speed of the second car is 40 miles per hour, and the speed of the first car is x + 12 = 52 miles per hour.

Step-by-step explanation:

Answer:

Let x = speed of first car x + 12 = speed of second

Step-by-step explanation:

Answer:

3.3.1 Approximations for trigonometric functions. 3.3.2 ... 7.2.1 Consistency for three simultaneous linear equations in two unknowns ... x : y : z =5:4:3;.

On  the number line, the points that represent the values of x in - 1 < x < 3 are:

The expression given is - 1 < x < 3 which means that x is any whole number that is greater than - 1 but less than 3. This means that on the number line given, the value of x would be any number that is more than - 1 but less than 3.

These numbers will include 0, 1, and 2. The line should therefore begin at -1 and go to 3 to show the values of x in - 1 < x < 3 . The line to use should be the open circle which shows greater than or less than

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We can calculate the value of {x} as 63.2°.

There are six major trigonometric functions as -

We can write the relation between them as -

Also the following trigonometry relations hold true -

Given is that -

cos{x} = 0.4505

We have -

cos{x} = 0.4505

{x} = cos⁻¹(0.4505)

{x} = cos⁻¹{cos(63.2)}

{x} = 63.2°

Therefore, we can calculate the value of {x} as 63.2°.

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By similarity ratio of triangles, we have;

ΔPQR ∼ ΔPSR ∼ ΔRSQ

PS/PR = PR/PQ

PQ/RQ = RQ/SQ

Similar triangles are defined as triangles that have the same shape, but their sizes may vary. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

Thus, applying the concept of similar triangle ratio we can say that;

ΔPQR ∼ ΔPSR ∼ ΔRSQ

Similarly, by similarity ratio of triangles, we have;

PS/PR = PR/PQ

We also have;

PQ/RQ = RQ/SQ

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Therefore , the solution of the given problem of trigonometry comes out to be  it is roughly 5193 feet from pier A to the coral reef.

Relationships between mathematics but also cubic splines When the different fields were combined, astrophysics is thought to have been born in the third century BC. With the aid of exact angle mathematical techniques, many metric issues can be resolved or the outcomes of calculating them can be ascertained. The study of the six fundamental trigonometric formulas is known as trigonometry.

Here,

The north-south distance from pier A to the coral reef, x sin(31°37′), is the other side of the triangle (relative to angle ). The east-west path from dock A to the edge of the coral reef, which is equal to x cos(31°37′), forms the opposite side of the triangle. We're looking for x's number.

The Pythagorean theory allows us to write:

=> (x cos(31°37′) + 2599) + (x sin(61°37′)) = (x sin(31°37′))

By condensing and rearrangeing, we obtain:

=> x² - 5198x + 6790401 = 0

We can use the quadratic formula to answer the following quadratic equation:

=> x = (-(-5198) ± √(-5198)^2 - 4(1)(6790401))) / (2(1))

=> x = (5198 + √(26857604)) / 2

=> x = (5198 ± 5188) / 2

=> x = 5193 or x = 5

The negative answer x = 5 does not make logic in this situation because x is a measure of distance. Consequently, it is roughly 5193 feet from pier A to the coral reef.

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The required value of x for the given figure is 2√7 units.

A right triangle, also known as a right-angled triangle, right-perpendicular triangle, orthogonal triangle, or formerly rectangled triangle, is a triangle with one right angle, or two perpendicular edges. The foundation of geometry is the relationship between the sides and other angles of the right triangle.

In the given figure we have two triangle big and small triangle.

To find the value of x;

first we use Pythagoras theorem in big triangle.

So; let base of big triangle is b

9² = 7² + b²

b² = 81 - 49

b = √32

Now in small triangle;

(√32)² = x² + 2²

32 = x² + 4

x² = 28

x = √28

x = 2√7 units.

Thus, required value of x is 2√7 units.

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Answer:

Percent rate of change per unit, t ≈ 10.99%

Step-by-step explanation:

The percent rate of change per unit, t, for a function is equal to the derivative of the function with respect to t, divided by the function value at t.

For the given function g(t) = 200(1.12)^t, the derivative with respect to t is:

g'(t) = ln(1.12) * 200(1.12)^t

The function value at t is:

g(t) = 200(1.12)^t

Therefore, the percent rate of change per unit, t, is:

g'(t) / g(t) = [ln(1.12) * 200(1.12)^t] / [200(1.12)^t] = ln(1.12) ≈ 0.1099

Multiplying by 100 to express the answer as a percentage, rounded to the nearest tenth of a percent, we get:

Percent rate of change per unit, t ≈ 10.99%

The expression when evaluated is 4√2x

From the question, we have the following parameters that can be used in our computation:

√32x

Express 32 as the product of 16 and 2

So, we have

√32x = √(16 * 2x)

Take the LCM of 16

So, we have the following representation

√32x = 4√2x

Hence, the solution is 4√2x

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72 items were checked to work correctly.

What is the solution of algebraic expression?

Finding the value of the variable is part of solving an algebraic statement. We must do certain algebraic operations to determine the value of one of the variables in the linear equations before we can substitute that value to get the value of another variable.

Then the percentage of defective items in the checked batch is x + 3%.

We know that out of 300 items, this percentage represents the number of defective items. So we can write:

(x + 3%) * 300 = number of defective items

Simplifying this equation, we get:

3x + 9 = number of defective items

number of items checked - number of defective items = number of items checked to work correctly

Substituting the equation we derived above for the number of defective items, we get:

300 - (3x + 9) = number of items checked to work correctly

Simplifying this equation, we get:

291 - 3x = number of items checked to work correctly

3x = number of defective items in a batch of 300 items with the normal percentage of defects

Substituting this expression for the number of defective items in our previous equation, we get:

291 - 3x = number of items checked to work correctly

Now we can solve for x:

3x = number of defective items in a batch of 300 items with the normal percentage of defects = x/100 * 300

3x = 3 * (300 - number of items checked to work correctly - number of defective items)

3x = 900 - 3(3x + 9)

3x = 900 - 9x - 27

12x = 873

x = 73

So the normal percentage of defective items is 73%.

Now we can find the number of items checked to work correctly:

291 - 3x = 291 - 3(73) = 72

Therefore, 72 items were checked to work correctly.

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The correct statement regarding the limit of f(x) as x -> 1 is given as follows:

lim x -> 1 f(x) = -∞.

To obtain lateral limits, we need to take the limit of a function as it approaches a point from the left-hand side and the right-hand side separately.

If the two limits are equal, then this is the limit of the function, while if the lateral limits are different, then the limit of the function does not exist for the value of x.

From quadrant 2, the function approaches x = 1 on quadrant 4, hence the limit to the left is given as follows:

lim x -> 1^- f(x) = -∞.

A graph approaches x = 1 in quadrant 1, and curves down and approaches the x-axis in quadrant 1, hence the limit to the right is given as follows:

lim x -> 1^+ f(x) = -∞.

As the lateral limits are equal, the limit of the function is given as follows:

lim x -> 1 f(x) = -∞.

The problem describes the graph of the function.

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Answer:

Step-by-step explanation:

5*x+11=36

5*x=36-11

x=25/5

x=5

5 is the answer

Hey there!

Guide to follow

• Product = multiply/multiplication
• Sum = add/addition
• Quotient = divide/division
• Difference = subtract/subtraction

• Is = equal to/equivalent to

Question reads….
“The product of a 5 and a number plus 11 is 36”

We are not sure what the unknown number is, so we will label it as “w”

Your equation:

5 * w + 11 = 36

SOLVING for the answer to the equation:

5 * w + 11 = 36
5w + 11 = 36

SUBTRACT to both sides

5w + 11 - 11 = 36 - 11

SIMPLIFY it

5w = 25

DIVIDE 5 to BOTH SIDES

5w/5 = 25/5

SIMPLIFY it

w = 25/5

w = 5

Therefore, your unknown number should be: 5

Good luck on your assignment & enjoy your day!

~Amphitrite1040:)

Answer: I dont understand the answer that much but off of what i understand i think the answer is A

Step-by-step explanation: I am not sure but i think so

Answer:

x = 3

y = -10

Step-by-step explanation:

  5x + y = 5                                                                    y = 3x - 19

  5x + 3x - 19 = 5

  8x - 19 = 5

  8x = 24

   x = 3

Now put x in to solve for y

y = 3(3) - 19

y = 9 - 19

y = -10

Let's check

5(3) - 10 = 5

15 - 10 = 5

5 = 5

So, x = 3 and y = -10 is the correct answer.